Camera
Optics

Depth of Field

A camera is replete with examples of physics, ranging from
levers and other simple machines to the advanced electronics that
controls the lens and shutter in most modern cameras. The purpose
of this part of the web site, though, will be to deal with one
aspect of cameras, namely the phenomenon called "depth of field."

It seems that when we adjust the aperture down to a smaller
opening, the depth of field increases, meaning that objects from a
wider range of distances will appear focused.

We'll use ray diagrams to investigate this idea, and will try
to show how stopping a lens down (making its aperture smaller)
will lead directly to depth of field. In the diagrams which
follow, there is a lens and two objects. The nearer object is
solid while the further object is shaded. Notice where each forms
its image in our example, and then keep following along.

This is the basic setup showing the lens, the two objects, the
axis of the lens and the two focal points, one on each side.

We form the image of the solid object first. This is just a
basic ray diagram.

Then we form the image of the gray object. The two images don't
fall the same distance from the lens. This is shown in this
diagram:

For discussion sake, we make the two images the same size but
retain their respective distances from the lens. Now place a piece
of film where the solid image is located in order to obtain a
sharp image of it.

Follow light rays as they go from the shaded object and hit all
over the lens. The result is a "zone of confusion" in the plane of
the film. In other words, the gray image is blurry.

If we introduce an aperture which reduces the size of the lens
opening, notice how it also cuts down on the size of the zone of
confusion. Making the opening smaller reduces the amount of
spreading out the image does after it has become a sharp image.

And cutting down the aperture opening further reduces the zone
of confusion even more. At this point the two images, dark and
gray, are almost equally sharp.

The phenomenon of Depth of Field, thus, is due to a geometric
aspect. By reducing the size of the aperture, the angle that light
passes through the lens and then through the image is reduced. The
degree to which the image is fuzzy has gone down.

In point of fact, only objects at one particular distance are
clearly in focus. But if there is a small angle involved, objects
at other distances will appear to also be clearly focused.

PINHOLE CAMERA

A Pinhole Camera consists of a small aperture that light passes
through, a dark enclosure, and a piece of film. As shown in the
diagram below, an object placed in front of a Pinhole Camera forms
a clear image on the film without the need for a lens.

Because the aperture is so small, from each point on the
object, only one ray of light may pass through and then move on to
the film. Thus there is a one-to-one correspondence between points
on the film and points on the object.

If we look at the possibility of changing the distance from the
pinhole to the film, we see that we can affect the size of the
image of a given object. Compare the size of the images below at
three different image distances.

Now, if we only consider energy, the energy concentration in
the smaller image is higher, and thus a shorter time is needed to
expose the film. The concentration in the larger image is less, so
a longer time is needed. Photographers in the past developed a
good understanding of this relationship.

If by accident more than one pinhole is opened, multiple images
are formed, one from each pinhole. Such an occurrence is
diagrammed below.

A very interesting phenomenon occurs, however, if a lens of the
correct focal length is inserted between the pinholes and the
film. The multiple images become a single image. This is
dramatically shown in the PSSC film "Introduction to Optics." The
demonstration is shown below schematically.

The explanation for this phenomenon will await an updated
version of this stack. The author believes that a correct
explanation, though, will demonstrate good understanding of not
only pinhole cameras, but of image formation in general.

f-STOPS

The question we confront at this point is the unusual pattern
of aperture opening numbers. On an adjustable camera, the f-stops
come in a sequence like the following:

1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22

Why does this particular sequence exist, and what does it mean?
Is it related to physics?

In understanding this puzzle, we first note that the aperture
is the opening through which light enters the camera to expose the
film. The larger the opening, the more light energy per second
enters. Therefore, we would imagine that a larger number would
indicate a larger aperture.

In fact, a larger f-stop number indicates a smaller aperture.
f/22 is a much smaller opening than f/2.

One clue to the puzzle comes from the notation, f/22. With a
larger number in the denominator, a fraction has a smaller value.
Therefore there must be an inverse relationship of some sort here.

Another interesting clue comes from the relative sizes. From
1.4 to 2.0, from 2.0 to 2.8, from 8 to 11 the size of the number
increases approximately 1.4 times each step. This is almost
exactly the square root of 2. From 2.0 to 4.0 requires 2 steps to
double, while from 4.0 to 8.0 also requires 2 steps.

Therefore we can home in on an answer. The f-stop is a
reciprocal number, indicating 1 divided by some factor. As the
f-stop increases one "step", the denominator increases by the
square root of 2, or the fraction reduces by 1 over the square
root of 2.

This number refers to the diameter or the radius of the
aperture. If the radius is reduced by the square root of 2, the
area is reduced by a factor of 2. But this is indicated by
increasing the f-stop by 1.4 times!

Consult the chart below to see the relative change in light
energy admitted to the camera, assuming the camera is opened for
equal amounts of time.

f-Stop

Relative Value

Light Energy

1.4

1

256

2.0

1/2

128

2.8

1/4

64

4.0

1/8

32

5.6

1/16

16

8.0

1/32

8

11

1/64

4

16

1/128

2

22

1/256

1

The application of physics here involves the interplay between
aperture opening and shutter speed. Both are set up roughly into
factors of 2. Shutter speeds go 1/30, 1/60, 1/125, 1/250, etc.

By increasing the shutter speed say from 1/60 to 1/125, one
decreases the light by a factor of 2. To compensate, he/she opens
the shutter up by changing the f-stop one setting, say from f/8 to
f/5.6. Knowing these factors of 2 can be of great assistance in
estimating exposure settings for the serious photographer.